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I would like to do a graphic search specifically finding places where the articular noun hamartia is used in the singular, and verses that also contain a second usage of hamartia. I can easily generate the definite article and hamartia search, so how do I tell it to look for that same noun again?

As you noted, it's pretty easy to find the article and noun together. In the graphical search, I made sure the article and noun agreed, and I also added that they be at most two words apart (to allow for δε or μεν δε...).
Then you add αμαρτια again, but by requiring that it be "at least 1" word apart from the article/noun combination, it forces it to find another instance of the word in the verse.

Do note that you will get a different set of hits looking for αμαρτια followed by the article/noun